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Metamath Proof Explorer

Theorem 4atlem11a

Description: Lemma for 4at . Substitute U for Q . (Contributed by NM, 9-Jul-2012)

Ref Expression
Hypotheses 4at.l 
|- .<_ = ( le ` K )
4at.j 
|- .\/ = ( join ` K )
4at.a 
|- A = ( Atoms ` K )
Assertion 4atlem11a 
|- ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) -> ( Q .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) <-> ( ( P .\/ Q ) .\/ ( V .\/ W ) ) = ( ( P .\/ U ) .\/ ( V .\/ W ) ) ) )

Proof

Step Hyp Ref Expression
1 4at.l 
 |-  .<_ = ( le ` K )
2 4at.j 
 |-  .\/ = ( join ` K )
3 4at.a 
 |-  A = ( Atoms ` K )
4 simp11 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) -> K e. HL )
5 simp13 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) -> Q e. A )
6 simp21 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) -> U e. A )
7 4 hllatd 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) -> K e. Lat )
8 simp12 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) -> P e. A )
9 simp22 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) -> V e. A )
10 eqid 
 |-  ( Base ` K ) = ( Base ` K )
11 10 2 3 hlatjcl 
 |-  ( ( K e. HL /\ P e. A /\ V e. A ) -> ( P .\/ V ) e. ( Base ` K ) )
12 4 8 9 11 syl3anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) -> ( P .\/ V ) e. ( Base ` K ) )
13 simp23 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) -> W e. A )
14 10 3 atbase 
 |-  ( W e. A -> W e. ( Base ` K ) )
15 13 14 syl 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) -> W e. ( Base ` K ) )
16 10 2 latjcl 
 |-  ( ( K e. Lat /\ ( P .\/ V ) e. ( Base ` K ) /\ W e. ( Base ` K ) ) -> ( ( P .\/ V ) .\/ W ) e. ( Base ` K ) )
17 7 12 15 16 syl3anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) -> ( ( P .\/ V ) .\/ W ) e. ( Base ` K ) )
18 simp3 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) -> -. Q .<_ ( ( P .\/ V ) .\/ W ) )
19 10 1 2 3 hlexchb2 
 |-  ( ( K e. HL /\ ( Q e. A /\ U e. A /\ ( ( P .\/ V ) .\/ W ) e. ( Base ` K ) ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) -> ( Q .<_ ( U .\/ ( ( P .\/ V ) .\/ W ) ) <-> ( Q .\/ ( ( P .\/ V ) .\/ W ) ) = ( U .\/ ( ( P .\/ V ) .\/ W ) ) ) )
20 4 5 6 17 18 19 syl131anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) -> ( Q .<_ ( U .\/ ( ( P .\/ V ) .\/ W ) ) <-> ( Q .\/ ( ( P .\/ V ) .\/ W ) ) = ( U .\/ ( ( P .\/ V ) .\/ W ) ) ) )
21 1 2 3 4atlem4b 
 |-  ( ( ( K e. HL /\ P e. A /\ U e. A ) /\ ( V e. A /\ W e. A ) ) -> ( ( P .\/ U ) .\/ ( V .\/ W ) ) = ( U .\/ ( ( P .\/ V ) .\/ W ) ) )
22 4 8 6 9 13 21 syl32anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) -> ( ( P .\/ U ) .\/ ( V .\/ W ) ) = ( U .\/ ( ( P .\/ V ) .\/ W ) ) )
23 22 breq2d 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) -> ( Q .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) <-> Q .<_ ( U .\/ ( ( P .\/ V ) .\/ W ) ) ) )
24 1 2 3 4atlem4b 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( V e. A /\ W e. A ) ) -> ( ( P .\/ Q ) .\/ ( V .\/ W ) ) = ( Q .\/ ( ( P .\/ V ) .\/ W ) ) )
25 4 8 5 9 13 24 syl32anc 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) -> ( ( P .\/ Q ) .\/ ( V .\/ W ) ) = ( Q .\/ ( ( P .\/ V ) .\/ W ) ) )
26 25 22 eqeq12d 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) -> ( ( ( P .\/ Q ) .\/ ( V .\/ W ) ) = ( ( P .\/ U ) .\/ ( V .\/ W ) ) <-> ( Q .\/ ( ( P .\/ V ) .\/ W ) ) = ( U .\/ ( ( P .\/ V ) .\/ W ) ) ) )
27 20 23 26 3bitr4d 
 |-  ( ( ( K e. HL /\ P e. A /\ Q e. A ) /\ ( U e. A /\ V e. A /\ W e. A ) /\ -. Q .<_ ( ( P .\/ V ) .\/ W ) ) -> ( Q .<_ ( ( P .\/ U ) .\/ ( V .\/ W ) ) <-> ( ( P .\/ Q ) .\/ ( V .\/ W ) ) = ( ( P .\/ U ) .\/ ( V .\/ W ) ) ) )